Pedal Equation

Pedal Equation is a fascinating concept in geometry that relates a specific point on a curve to its tangent lines. In simple terms, the pedal equation describes the relationship between two key distances: the distance from a fixed point (known as the pedal point) to any point on a curve, and the perpendicular distance from this fixed point to the tangent of the curve at that point.

To better understand this, imagine you have a curve and a fixed point. The pedal equation helps you figure out how these distances change as you move along the curve. This relationship is crucial in understanding various properties of curves, such as their shape and how they behave when you apply certain transformations.

What is a Pedal Equation?

A pedal equation is understood as an equation such that through the distance from some fixed point referred to as the pedal point a function is determined which is proportional to the tangent of a curve. In the case of a curve in the plane, the pedal equation gives the relation between the distance of a point on the curve from a fixed point called the pedal point and the tangent at the point on the curve.

For a curve given in polar coordinates as r=f(θ), the pedal equation can be derived using the following relation:

p = rsin(ψ)

Here, ‘p’ stands for the perpendicular distance of the pedal point from a tangent passing through a given point on the curve, ‘r’ is the distance of the point on the curve from the origin, and ‘ψ’ is the angle that tangent makes with the radius vector drawn at that particular point. By expressing the tangential components of the curve in terms of the pedal point this equation helps to reveal certain geometric properties of the curve.

Deriving the Pedal Equation

Deriving the pedal equation involves several steps:

• Suppose that the curve is given by the polar equation r = f(θ).
• Tangent at that point can also be found by differentiating the polar equation to get dr/dθ.
• Pedal equation is obtained by projecting the radial distance onto the tangent and using trigonometric identities to relate r, dr/dθ, and p.
• Final form of the pedal equation is derived by eliminating unnecessary variables and expressing p purely in terms of r and θ.

For example, if the curve is a circle with the equation r = a cos(θ), the pedal equation can be derived as:

r = acos² (θ/2)

This derivation highlights the connections between the curve's geometry and its pedal properties.

Examples of Pedal Equations

Pedal equations can vary depending on the type of curve and the location of the pedal point. Below are some common types of pedal equations with their respective formulas:

Pedal Equation of a Circle

When the pedal point is at the center of the circle, the pedal equation is given by:

p = r \sin(\theta)

Where p is the distance from the pedal point to the tangent, r is the distance from the curve’s origin to the point on the curve, and θ is the angle between the radius vector and the tangent.

Pedal Equation of a Parabola

For a parabola y² = 4ax, with the pedal point at the focus, the pedal equation is:

p = \frac{4a \sin(\theta)}{(1 + \cos(\theta))^2}

Here, a is the distance from the vertex to the focus, θ is the angle, and p is the pedal distance.

Pedal Equation of an Ellipse

For an ellipse x²/a² + y²/b² = 1, the pedal equation with the pedal point at one of the foci is:

p = \frac{b^2 r^2}{a^2}

Where a and b are the semi-major and semi-minor axes, respectively, and r is the radial distance.

Pedal Equation of a Hyperbola

For a hyperbola x²/a² - y²/b² = 1, the pedal equation when the pedal point is at one of the foci is:

p = \frac{b^2 r^2}{a^2 \cosh^2(\theta)}

Here, a and b are the semi-major and semi-minor axes, and θ is the hyperbolic angle.

Pedal Equation of a Straight Line

For a straight line, the pedal equation is relatively simple. If the line is given in polar form as r = p sec(θ − α), where p is the perpendicular distance from the origin to the line and α is the angle of the perpendicular, the pedal equation is:

p = \frac{r}{\sin(\theta - \alpha)}

How to Solve Pedal Equation?

Solving pedal equations typically involves:

Step 1: Start with the curve's equation in either Cartesian or polar form.

Step 2: Use the appropriate pedal equation based on the curve's representation.

Step 3: Solve for p or r, depending on the form of the equation, by substituting known values and simplifying.

Step 4: Interpret the solution in the context of the curve's geometry and the position of the pedal point.

For example, if we start with the circle r = a cos(θ), the solution involves calculating the pedal point distance p and interpreting its significance about the circle.

Applications of Pedal Equations

• Optics: Used in the study of light paths and reflection properties.
• Physics: Helps analyze particle trajectories under central forces.
• Engineering: Applied in mechanical systems where motion paths are critical.
• Robotics: Essential for path planning algorithms in navigation systems.

Conclusion

Pedal Equations provides a real depth to the subject and they find their use in mathematics, physics and engineering applications and provide an insight into the geometric properties of curves in terms of pedal equations. The knowledge of how they are derived, solved and applied is essential to the students as well as the professionals. As mathematical tools advance, the study of pedal equations continues to evolve, opening new possibilities for research and practical application.

Read More,

• Differential Equation
• Cartesian Coordinate System
• Polar Coordinate System

Solved Examples

Question 1: Find the pedal equation of a circle with radius a, where the pedal point is at the center of the circle.

Solution:

Step 1: Start with the equation of the circle in Cartesian coordinates:

x² + y² = a²

Step 2: Convert the equation to polar coordinates, where r is the radial distance, and θ is the angle:

r² = a²

Step 3: Since the pedal point is at the center of the circle, the pedal equation is derived directly using the relation p = r sin(ψ). For a circle, r is constant and equal to the radius a, and ψ = θ. Therefore, the pedal equation simplifies to:

p = asin(θ)

Step 4: Recognize that for all points on the circle, the value of sin(θ) will vary, but the relationship between p and r remains consistent, giving us:

p = a sin(θ)

Thus, the pedal equation of the circle with a center at the origin is p = a sin(θ).

Question 2: Find the pedal equation of a parabola y² = 4ax, where the pedal point is at the focus of the parabola.

Solution:

Step 1: Start with the equation of the parabola in Cartesian coordinates:

y² = 4ax

Step 2: For a parabola, the distance from the focus to any point on the curve is given by r, and the distance from the directrix to the same point is also equal to r. Convert the equation to polar coordinates by expressing x and y in terms of r and θ:

r = \frac{4a}{1 + \cos(\theta)}

Step 3: The pedal equation with the pedal point at the focus is derived using the standard relation:

p = \frac{r \sin(\theta)}{1 + \cos(\theta)}

Given the relation for r, the pedal equation becomes:

p = \frac{4a \sin(\theta)}{(1 + \cos(\theta))^2}

Step 4: Simplify the equation by eliminating the trigonometric terms where possible:

p = \frac{4a \sin(\theta)}{1 + 2\cos(\theta) + \cos^2(\theta)}

Step 5: Recognize that the equation can be further simplified if necessary, but it already represents the relationship between the perpendicular distance p from the pedal point (focus) to the tangent at the curve for the given parabola.

Thus, the pedal equation of the parabola y² = 4ax with the pedal point at the focus is:

p = \frac{4a \sin(\theta)}{(1 + \cos(\theta))^2}

Practice Problems on Pedal Equation

Problem 1: Find the pedal equation of the circle r = a cos⁡(θ).

Problem 2: Derive the pedal equation for a parabola where the focus is at the origin.

Problem 3: Given the pedal equation p² = c²(1 + cos⁡(2θ)), determine the corresponding curve.

Problem 4: Find the pedal equation of the ellipse given by x²/a² + y²/b² = 1.

Problem 5: Calculate the pedal equation of a hyperbola where the semi-major axis is 'a' and the semi-minor axis is 'b'.

FAQs on Pedal Equation

What is the purpose of the pedal equation?

The pedal equation describes the relationship between the radius of curvature and the distance from a given point (called the pedal point) to a curve.

What is the main function of pedals?

Pedals refer to points on a curve, and the pedal equation helps in studying the curve's geometry concerning a fixed point.

What is the pedal form of a circle?

The pedal equation of a circle is indicated by the formula p = r cos θ, where r stands for the radius, p stands for the perpendicular distance from the pedal point, and θ stands for the angle.

How to solve the pedal equation?

To solve a pedal equation, rewrite the curve equation regarding distance from the pedal point and differentiate it in terms of angle or arc length.

T

tabass8o64

Follow

Improve

Article Tags :

• Mathematics
• Engineering Mathematics
• Maths
• Calculus